diff --git a/crates/fj-kernel/src/algorithms/approx/edge.rs b/crates/fj-kernel/src/algorithms/approx/edge.rs
index 906a059c3..08af8a8a2 100644
--- a/crates/fj-kernel/src/algorithms/approx/edge.rs
+++ b/crates/fj-kernel/src/algorithms/approx/edge.rs
@@ -8,7 +8,7 @@
 use std::collections::BTreeMap;
 
 use crate::{
-    geometry::path::{GlobalPath, SurfacePath},
+    geometry::curve::{Curve, GlobalPath},
     objects::{GlobalEdge, HalfEdge, Surface},
     storage::{Handle, ObjectId},
 };
@@ -91,7 +91,7 @@ impl HalfEdgeApprox {
 }
 
 fn approx_edge(
-    curve: &SurfacePath,
+    curve: &Curve,
     surface: &Surface,
     range: RangeOnPath,
     tolerance: impl Into<Tolerance>,
@@ -103,12 +103,12 @@ fn approx_edge(
     // `GlobalPath` grow APIs that are better suited to implementing this code
     // in a more abstract way.
     let points = match (curve, surface.geometry().u) {
-        (SurfacePath::Circle(_), GlobalPath::Circle(_)) => {
+        (Curve::Circle(_), GlobalPath::Circle(_)) => {
             todo!(
                 "Approximating a circle on a curved surface not supported yet."
             )
         }
-        (SurfacePath::Circle(_), GlobalPath::Line(_)) => {
+        (Curve::Circle(_), GlobalPath::Line(_)) => {
             (curve, range)
                 .approx_with_cache(tolerance, &mut ())
                 .into_iter()
@@ -135,7 +135,7 @@ fn approx_edge(
                 })
                 .collect()
         }
-        (SurfacePath::Line(line), _) => {
+        (Curve::Line(line), _) => {
             let range_u =
                 RangeOnPath::from(range.boundary.map(|point_curve| {
                     [curve.point_from_path_coords(point_curve).u]
@@ -232,7 +232,7 @@ mod tests {
     use crate::{
         algorithms::approx::{path::RangeOnPath, Approx, ApproxPoint},
         builder::{HalfEdgeBuilder, SurfaceBuilder},
-        geometry::path::GlobalPath,
+        geometry::curve::GlobalPath,
         insert::Insert,
         partial::{PartialHalfEdge, PartialObject, PartialSurface},
         services::Services,
diff --git a/crates/fj-kernel/src/algorithms/approx/path.rs b/crates/fj-kernel/src/algorithms/approx/path.rs
index 0c7c1d579..aa45a710b 100644
--- a/crates/fj-kernel/src/algorithms/approx/path.rs
+++ b/crates/fj-kernel/src/algorithms/approx/path.rs
@@ -32,11 +32,11 @@ use std::iter;
 
 use fj_math::{Circle, Point, Scalar, Sign};
 
-use crate::geometry::path::{GlobalPath, SurfacePath};
+use crate::geometry::curve::{Curve, GlobalPath};
 
 use super::{Approx, Tolerance};
 
-impl Approx for (&SurfacePath, RangeOnPath) {
+impl Approx for (&Curve, RangeOnPath) {
     type Approximation = Vec<(Point<1>, Point<2>)>;
     type Cache = ();
 
@@ -48,10 +48,10 @@ impl Approx for (&SurfacePath, RangeOnPath) {
         let (path, range) = self;
 
         match path {
-            SurfacePath::Circle(circle) => {
+            Curve::Circle(circle) => {
                 approx_circle(circle, range, tolerance.into())
             }
-            SurfacePath::Line(_) => vec![],
+            Curve::Line(_) => vec![],
         }
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
index 495718be1..8fea643fa 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
@@ -1,6 +1,6 @@
 use fj_math::{Point, Segment};
 
-use crate::{geometry::path::SurfacePath, objects::HalfEdge};
+use crate::{geometry::curve::Curve, objects::HalfEdge};
 
 use super::LineSegmentIntersection;
 
@@ -28,15 +28,15 @@ impl CurveEdgeIntersection {
     /// Currently, only intersections between lines and line segments can be
     /// computed. Panics, if a different type of curve or [`HalfEdge`] is
     /// passed.
-    pub fn compute(curve: &SurfacePath, half_edge: &HalfEdge) -> Option<Self> {
+    pub fn compute(curve: &Curve, half_edge: &HalfEdge) -> Option<Self> {
         let curve_as_line = match curve {
-            SurfacePath::Line(line) => line,
+            Curve::Line(line) => line,
             _ => todo!("Curve-edge intersection only supports lines"),
         };
 
         let edge_as_segment = {
             let edge_curve_as_line = match half_edge.curve() {
-                SurfacePath::Line(line) => line,
+                Curve::Line(line) => line,
                 _ => {
                     todo!("Curve-edge intersection only supports line segments")
                 }
@@ -73,7 +73,7 @@ mod tests {
 
     use crate::{
         builder::HalfEdgeBuilder,
-        geometry::path::SurfacePath,
+        geometry::curve::Curve,
         partial::{PartialHalfEdge, PartialObject},
         services::Services,
     };
@@ -85,7 +85,7 @@ mod tests {
         let mut services = Services::new();
 
         let surface = services.objects.surfaces.xy_plane();
-        let curve = SurfacePath::u_axis();
+        let curve = Curve::u_axis();
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
             half_edge.update_as_line_segment_from_points([[1., -1.], [1., 1.]]);
@@ -109,7 +109,7 @@ mod tests {
         let mut services = Services::new();
 
         let surface = services.objects.surfaces.xy_plane();
-        let curve = SurfacePath::u_axis();
+        let curve = Curve::u_axis();
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
             half_edge
@@ -134,7 +134,7 @@ mod tests {
         let mut services = Services::new();
 
         let surface = services.objects.surfaces.xy_plane();
-        let curve = SurfacePath::u_axis();
+        let curve = Curve::u_axis();
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
             half_edge
@@ -154,7 +154,7 @@ mod tests {
         let mut services = Services::new();
 
         let surface = services.objects.surfaces.xy_plane();
-        let curve = SurfacePath::u_axis();
+        let curve = Curve::u_axis();
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
             half_edge.update_as_line_segment_from_points([[-1., 0.], [1., 0.]]);
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
index c6acad738..61020a75a 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
@@ -3,7 +3,7 @@ use std::vec;
 use fj_interop::ext::SliceExt;
 use fj_math::Point;
 
-use crate::{geometry::path::SurfacePath, objects::Face};
+use crate::{geometry::curve::Curve, objects::Face};
 
 use super::CurveEdgeIntersection;
 
@@ -28,7 +28,7 @@ impl CurveFaceIntersection {
     }
 
     /// Compute the intersection
-    pub fn compute(curve: &SurfacePath, face: &Face) -> Self {
+    pub fn compute(curve: &Curve, face: &Face) -> Self {
         let half_edges = face.all_cycles().flat_map(|cycle| cycle.half_edges());
 
         let mut intersections = Vec::new();
@@ -151,7 +151,7 @@ where
 mod tests {
     use crate::{
         builder::{CycleBuilder, FaceBuilder},
-        geometry::path::SurfacePath,
+        geometry::curve::Curve,
         partial::{Partial, PartialFace, PartialObject},
         services::Services,
     };
@@ -162,7 +162,7 @@ mod tests {
     fn compute() {
         let mut services = Services::new();
 
-        let (curve, _) = SurfacePath::line_from_points([[-3., 0.], [-2., 0.]]);
+        let (curve, _) = Curve::line_from_points([[-3., 0.], [-2., 0.]]);
 
         #[rustfmt::skip]
         let exterior = [
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_face.rs b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
index 96e2f35a8..8ef66f4c5 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
@@ -1,7 +1,7 @@
 use fj_interop::ext::ArrayExt;
 use iter_fixed::IntoIteratorFixed;
 
-use crate::{geometry::path::SurfacePath, objects::Face};
+use crate::{geometry::curve::Curve, objects::Face};
 
 use super::{CurveFaceIntersection, SurfaceSurfaceIntersection};
 
@@ -14,7 +14,7 @@ pub struct FaceFaceIntersection {
     /// representation of the intersection on the respective face's surface.
     ///
     /// They both represent the same global curve.
-    pub intersection_curves: [SurfacePath; 2],
+    pub intersection_curves: [Curve; 2],
 
     /// The interval of this intersection, in curve coordinates
     ///
@@ -63,7 +63,7 @@ mod tests {
     use crate::{
         algorithms::intersect::CurveFaceIntersection,
         builder::CycleBuilder,
-        geometry::path::SurfacePath,
+        geometry::curve::Curve,
         partial::{Partial, PartialFace, PartialObject},
         services::Services,
     };
@@ -128,7 +128,7 @@ mod tests {
         let intersection = FaceFaceIntersection::compute([&a, &b]);
 
         let expected_curves = surfaces.map(|_| {
-            let (path, _) = SurfacePath::line_from_points([[0., 0.], [1., 0.]]);
+            let (path, _) = Curve::line_from_points([[0., 0.], [1., 0.]]);
             path
         });
         let expected_intervals =
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs b/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
index faf7c4c47..ef753c041 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
@@ -4,7 +4,7 @@ use fj_math::Segment;
 
 use crate::{
     algorithms::intersect::{HorizontalRayToTheRight, Intersect},
-    geometry::path::SurfacePath,
+    geometry::curve::Curve,
     objects::HalfEdge,
     storage::Handle,
 };
@@ -18,8 +18,8 @@ impl Intersect for (&HorizontalRayToTheRight<2>, &Handle<HalfEdge>) {
         let (ray, edge) = self;
 
         let line = match edge.curve() {
-            SurfacePath::Line(line) => line,
-            SurfacePath::Circle(_) => {
+            Curve::Line(line) => line,
+            Curve::Circle(_) => {
                 todo!("Casting rays against circles is not supported yet")
             }
         };
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
index 4d90f15c3..df4937d90 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
@@ -4,7 +4,7 @@ use fj_math::{Plane, Point, Scalar};
 
 use crate::{
     algorithms::intersect::face_point::FacePointIntersection,
-    geometry::path::GlobalPath,
+    geometry::curve::GlobalPath,
     objects::{Face, HalfEdge, SurfaceVertex},
     storage::Handle,
 };
diff --git a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
index b6927ff1f..5dfbef44b 100644
--- a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
@@ -1,7 +1,7 @@
 use fj_math::{Line, Plane, Point, Scalar};
 
 use crate::{
-    geometry::path::{GlobalPath, SurfacePath},
+    geometry::curve::{Curve, GlobalPath},
     objects::Surface,
     storage::Handle,
 };
@@ -10,7 +10,7 @@ use crate::{
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct SurfaceSurfaceIntersection {
     /// The intersection curves
-    pub intersection_curves: [SurfacePath; 2],
+    pub intersection_curves: [Curve; 2],
 }
 
 impl SurfaceSurfaceIntersection {
@@ -48,8 +48,7 @@ impl SurfaceSurfaceIntersection {
 
         let line = Line::from_origin_and_direction(origin, direction);
 
-        let curves =
-            planes.map(|plane| SurfacePath::Line(plane.project_line(&line)));
+        let curves = planes.map(|plane| Curve::Line(plane.project_line(&line)));
 
         Some(Self {
             intersection_curves: curves,
@@ -76,7 +75,7 @@ mod tests {
     use pretty_assertions::assert_eq;
 
     use crate::{
-        algorithms::transform::TransformObject, geometry::path::SurfacePath,
+        algorithms::transform::TransformObject, geometry::curve::Curve,
         services::Services,
     };
 
@@ -101,8 +100,8 @@ mod tests {
             None,
         );
 
-        let expected_xy = SurfacePath::u_axis();
-        let expected_xz = SurfacePath::u_axis();
+        let expected_xy = Curve::u_axis();
+        let expected_xz = Curve::u_axis();
 
         assert_eq!(
             SurfaceSurfaceIntersection::compute([xy, xz],),
diff --git a/crates/fj-kernel/src/algorithms/sweep/curve.rs b/crates/fj-kernel/src/algorithms/sweep/curve.rs
index 6eaed3c74..6991871e5 100644
--- a/crates/fj-kernel/src/algorithms/sweep/curve.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/curve.rs
@@ -2,7 +2,7 @@ use fj_math::{Circle, Line, Vector};
 
 use crate::{
     builder::SurfaceBuilder,
-    geometry::path::{GlobalPath, SurfacePath},
+    geometry::curve::{Curve, GlobalPath},
     insert::Insert,
     objects::{Objects, Surface},
     partial::{PartialObject, PartialSurface},
@@ -12,7 +12,7 @@ use crate::{
 
 use super::{Sweep, SweepCache};
 
-impl Sweep for (SurfacePath, &Surface) {
+impl Sweep for (Curve, &Surface) {
     type Swept = Handle<Surface>;
 
     fn sweep_with_cache(
@@ -48,7 +48,7 @@ impl Sweep for (SurfacePath, &Surface) {
         }
 
         let u = match curve {
-            SurfacePath::Circle(circle) => {
+            Curve::Circle(circle) => {
                 let center = surface
                     .geometry()
                     .point_from_surface_coords(circle.center());
@@ -61,7 +61,7 @@ impl Sweep for (SurfacePath, &Surface) {
 
                 GlobalPath::Circle(circle)
             }
-            SurfacePath::Line(line) => {
+            Curve::Line(line) => {
                 let origin =
                     surface.geometry().point_from_surface_coords(line.origin());
                 let direction = surface
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index b061ddb68..b2a5035cc 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -6,7 +6,7 @@ use fj_math::{Scalar, Vector};
 use crate::{
     algorithms::{reverse::Reverse, transform::TransformObject},
     builder::{CycleBuilder, FaceBuilder},
-    geometry::path::GlobalPath,
+    geometry::curve::GlobalPath,
     insert::Insert,
     objects::{Face, Objects, Shell},
     partial::{Partial, PartialFace, PartialObject, PartialShell},
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index 0ed9d4e35..58dca3b83 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -3,11 +3,11 @@ use fj_math::{Point, Scalar};
 
 use crate::{
     geometry::{
-        path::{GlobalPath, SurfacePath},
+        curve::{Curve, GlobalPath},
         surface::SurfaceGeometry,
     },
     objects::{GlobalEdge, HalfEdge},
-    partial::{MaybeSurfacePath, Partial, PartialGlobalEdge, PartialHalfEdge},
+    partial::{MaybeCurve, Partial, PartialGlobalEdge, PartialHalfEdge},
 };
 
 /// Builder API for [`PartialHalfEdge`]
@@ -15,18 +15,18 @@ pub trait HalfEdgeBuilder {
     /// Update partial half-edge to represent the u-axis of the surface it is on
     ///
     /// Returns the updated path.
-    fn update_as_u_axis(&mut self) -> SurfacePath;
+    fn update_as_u_axis(&mut self) -> Curve;
 
     /// Update partial curve to represent the v-axis of the surface it is on
     ///
     /// Returns the updated path.
-    fn update_as_v_axis(&mut self) -> SurfacePath;
+    fn update_as_v_axis(&mut self) -> Curve;
 
     /// Update partial half-edge to be a circle, from the given radius
     fn update_as_circle_from_radius(
         &mut self,
         radius: impl Into<Scalar>,
-    ) -> SurfacePath;
+    ) -> Curve;
 
     /// Update partial half-edge to be an arc, spanning the given angle in
     /// radians
@@ -40,10 +40,10 @@ pub trait HalfEdgeBuilder {
     fn update_as_line_segment_from_points(
         &mut self,
         points: [impl Into<Point<2>>; 2],
-    ) -> SurfacePath;
+    ) -> Curve;
 
     /// Update partial half-edge to be a line segment
-    fn update_as_line_segment(&mut self) -> SurfacePath;
+    fn update_as_line_segment(&mut self) -> Curve;
 
     /// Infer the global form of the half-edge
     ///
@@ -74,14 +74,14 @@ pub trait HalfEdgeBuilder {
 }
 
 impl HalfEdgeBuilder for PartialHalfEdge {
-    fn update_as_u_axis(&mut self) -> SurfacePath {
-        let path = SurfacePath::u_axis();
+    fn update_as_u_axis(&mut self) -> Curve {
+        let path = Curve::u_axis();
         self.curve = Some(path.into());
         path
     }
 
-    fn update_as_v_axis(&mut self) -> SurfacePath {
-        let path = SurfacePath::v_axis();
+    fn update_as_v_axis(&mut self) -> Curve {
+        let path = Curve::v_axis();
         self.curve = Some(path.into());
         path
     }
@@ -89,8 +89,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
     fn update_as_circle_from_radius(
         &mut self,
         radius: impl Into<Scalar>,
-    ) -> SurfacePath {
-        let path = SurfacePath::circle_from_radius(radius);
+    ) -> Curve {
+        let path = Curve::circle_from_radius(radius);
         self.curve = Some(path.into());
 
         let [a_curve, b_curve] =
@@ -131,8 +131,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
 
         let arc = fj_math::Arc::from_endpoints_and_angle(start, end, angle_rad);
 
-        let path =
-            SurfacePath::circle_from_center_and_radius(arc.center, arc.radius);
+        let path = Curve::circle_from_center_and_radius(arc.center, arc.radius);
         self.curve = Some(path.into());
 
         let [a_curve, b_curve] =
@@ -152,7 +151,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
     fn update_as_line_segment_from_points(
         &mut self,
         points: [impl Into<Point<2>>; 2],
-    ) -> SurfacePath {
+    ) -> Curve {
         for (vertex, point) in self.vertices.each_mut_ext().zip_ext(points) {
             let mut surface_form = vertex.1.write();
             surface_form.position = Some(point.into());
@@ -161,7 +160,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         self.update_as_line_segment()
     }
 
-    fn update_as_line_segment(&mut self) -> SurfacePath {
+    fn update_as_line_segment(&mut self) -> Curve {
         let boundary = self.vertices.each_ref_ext().map(|vertex| vertex.0);
         let points_surface = self.vertices.each_ref_ext().map(|vertex| {
             vertex
@@ -174,12 +173,12 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         let path = if let [Some(start), Some(end)] = boundary {
             let points = [start, end].zip_ext(points_surface);
 
-            let path = SurfacePath::from_points_with_line_coords(points);
+            let path = Curve::from_points_with_line_coords(points);
             self.curve = Some(path.into());
 
             path
         } else {
-            let (path, _) = SurfacePath::line_from_points(points_surface);
+            let (path, _) = Curve::line_from_points(points_surface);
             self.curve = Some(path.into());
 
             for (vertex, position) in
@@ -212,7 +211,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         let path = self
             .curve
             .expect("Can't infer vertex positions without curve");
-        let MaybeSurfacePath::Defined(path) = path else {
+        let MaybeCurve::Defined(path) = path else {
             panic!("Can't infer vertex positions with undefined path");
         };
 
@@ -263,8 +262,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
                     // represented using our current curve/surface
                     // representation.
                     match path {
-                        MaybeSurfacePath::Defined(SurfacePath::Line(_))
-                        | MaybeSurfacePath::UndefinedLine => {
+                        MaybeCurve::Defined(Curve::Line(_))
+                        | MaybeCurve::UndefinedLine => {
                             // We're dealing with a line on a rounded surface.
                             //
                             // Based on the current uses of this method, we can
@@ -291,7 +290,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
                             // I hope that I'll come up with a better curve/
                             // surface representation before this becomes a
                             // problem.
-                            Some(MaybeSurfacePath::UndefinedCircle {
+                            Some(MaybeCurve::UndefinedCircle {
                                 radius: circle.radius(),
                             })
                         }
@@ -308,11 +307,11 @@ impl HalfEdgeBuilder for PartialHalfEdge {
                 GlobalPath::Line(_) => {
                     // The other edge is defined on a plane.
                     match path {
-                        MaybeSurfacePath::Defined(SurfacePath::Line(_))
-                        | MaybeSurfacePath::UndefinedLine => {
+                        MaybeCurve::Defined(Curve::Line(_))
+                        | MaybeCurve::UndefinedLine => {
                             // The other edge is a line segment on a plane. That
                             // means our edge must be a line segment too.
-                            Some(MaybeSurfacePath::UndefinedLine)
+                            Some(MaybeCurve::UndefinedLine)
                         }
                         _ => {
                             // The other edge is a circle or arc on a plane. I'm
diff --git a/crates/fj-kernel/src/builder/face.rs b/crates/fj-kernel/src/builder/face.rs
index 1a6a1f520..1f76547dc 100644
--- a/crates/fj-kernel/src/builder/face.rs
+++ b/crates/fj-kernel/src/builder/face.rs
@@ -3,9 +3,9 @@ use std::{array, collections::VecDeque};
 use fj_interop::ext::ArrayExt;
 
 use crate::{
-    geometry::path::SurfacePath,
+    geometry::curve::Curve,
     objects::{Cycle, Surface},
-    partial::{MaybeSurfacePath, Partial, PartialFace},
+    partial::{MaybeCurve, Partial, PartialFace},
 };
 
 use super::SurfaceBuilder;
@@ -100,13 +100,13 @@ impl FaceBuilder for PartialFace {
 
             if let Some(path) = &mut curve {
                 match path {
-                    MaybeSurfacePath::Defined(_) => {
+                    MaybeCurve::Defined(_) => {
                         // Path is already defined. Nothing to infer.
                     }
-                    MaybeSurfacePath::UndefinedCircle { .. } => todo!(
+                    MaybeCurve::UndefinedCircle { .. } => todo!(
                         "Inferring undefined circles is not supported yet"
                     ),
-                    MaybeSurfacePath::UndefinedLine => {
+                    MaybeCurve::UndefinedLine => {
                         let points_surface =
                             half_edge.vertices.each_ref_ext().map(|vertex| {
                                 vertex.1.read().position.expect(
@@ -114,9 +114,9 @@ impl FaceBuilder for PartialFace {
                                 )
                             });
                         let (line, points_curve) =
-                            SurfacePath::line_from_points(points_surface);
+                            Curve::line_from_points(points_surface);
 
-                        *path = MaybeSurfacePath::Defined(line);
+                        *path = MaybeCurve::Defined(line);
                         for (vertex, point) in half_edge
                             .vertices
                             .each_mut_ext()
diff --git a/crates/fj-kernel/src/builder/surface.rs b/crates/fj-kernel/src/builder/surface.rs
index 1d5e23ee7..a4a0181f4 100644
--- a/crates/fj-kernel/src/builder/surface.rs
+++ b/crates/fj-kernel/src/builder/surface.rs
@@ -1,7 +1,7 @@
 use fj_math::{Point, Scalar, Vector};
 
 use crate::{
-    geometry::{path::GlobalPath, surface::SurfaceGeometry},
+    geometry::{curve::GlobalPath, surface::SurfaceGeometry},
     partial::PartialSurface,
 };
 
diff --git a/crates/fj-kernel/src/geometry/path.rs b/crates/fj-kernel/src/geometry/curve.rs
similarity index 98%
rename from crates/fj-kernel/src/geometry/path.rs
rename to crates/fj-kernel/src/geometry/curve.rs
index 6ccd18ae2..fe4cb7d22 100644
--- a/crates/fj-kernel/src/geometry/path.rs
+++ b/crates/fj-kernel/src/geometry/curve.rs
@@ -1,12 +1,12 @@
 //! Paths through 2D and 3D space
 //!
-//! See [`SurfacePath`] and [`GlobalPath`].
+//! See [`Curve`] and [`GlobalPath`].
 
 use fj_math::{Circle, Line, Point, Scalar, Transform, Vector};
 
 /// A path through surface (2D) space
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
-pub enum SurfacePath {
+pub enum Curve {
     /// A circle
     Circle(Circle<2>),
 
@@ -14,7 +14,7 @@ pub enum SurfacePath {
     Line(Line<2>),
 }
 
-impl SurfacePath {
+impl Curve {
     /// Build a circle from the given radius
     pub fn circle_from_radius(radius: impl Into<Scalar>) -> Self {
         Self::circle_from_center_and_radius(Point::origin(), radius)
diff --git a/crates/fj-kernel/src/geometry/mod.rs b/crates/fj-kernel/src/geometry/mod.rs
index 654a3c108..af0cf08f9 100644
--- a/crates/fj-kernel/src/geometry/mod.rs
+++ b/crates/fj-kernel/src/geometry/mod.rs
@@ -1,4 +1,4 @@
 //! Types that are tied to objects, but aren't objects themselves
 
-pub mod path;
+pub mod curve;
 pub mod surface;
diff --git a/crates/fj-kernel/src/geometry/surface.rs b/crates/fj-kernel/src/geometry/surface.rs
index 612387e1d..bcdd27875 100644
--- a/crates/fj-kernel/src/geometry/surface.rs
+++ b/crates/fj-kernel/src/geometry/surface.rs
@@ -2,7 +2,7 @@
 
 use fj_math::{Line, Plane, Point, Transform, Vector};
 
-use super::path::GlobalPath;
+use super::curve::GlobalPath;
 
 /// The geometry that defines a surface
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
@@ -64,7 +64,7 @@ mod tests {
     use fj_math::{Line, Point, Vector};
     use pretty_assertions::assert_eq;
 
-    use crate::geometry::{path::GlobalPath, surface::SurfaceGeometry};
+    use crate::geometry::{curve::GlobalPath, surface::SurfaceGeometry};
 
     #[test]
     fn point_from_surface_coords() {
diff --git a/crates/fj-kernel/src/objects/full/cycle.rs b/crates/fj-kernel/src/objects/full/cycle.rs
index 99a83b67a..be42f3fda 100644
--- a/crates/fj-kernel/src/objects/full/cycle.rs
+++ b/crates/fj-kernel/src/objects/full/cycle.rs
@@ -3,7 +3,7 @@ use std::slice;
 use fj_interop::ext::SliceExt;
 use fj_math::{Scalar, Winding};
 
-use crate::{geometry::path::SurfacePath, objects::HalfEdge, storage::Handle};
+use crate::{geometry::curve::Curve, objects::HalfEdge, storage::Handle};
 
 /// A cycle of connected half-edges
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
@@ -56,8 +56,8 @@ impl Cycle {
             let edge_direction_positive = a < b;
 
             let circle = match first.curve() {
-                SurfacePath::Circle(circle) => circle,
-                SurfacePath::Line(_) => unreachable!(
+                Curve::Circle(circle) => circle,
+                Curve::Line(_) => unreachable!(
                     "Invalid cycle: less than 3 edges, but not all are circles"
                 ),
             };
diff --git a/crates/fj-kernel/src/objects/full/edge.rs b/crates/fj-kernel/src/objects/full/edge.rs
index 1245d702a..215641b3c 100644
--- a/crates/fj-kernel/src/objects/full/edge.rs
+++ b/crates/fj-kernel/src/objects/full/edge.rs
@@ -2,7 +2,7 @@ use fj_interop::ext::ArrayExt;
 use fj_math::Point;
 
 use crate::{
-    geometry::path::SurfacePath,
+    geometry::curve::Curve,
     objects::{GlobalVertex, SurfaceVertex},
     storage::Handle,
 };
@@ -45,7 +45,7 @@ use crate::{
 /// <https://github.com/hannobraun/Fornjot/issues/1608>
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct HalfEdge {
-    curve: SurfacePath,
+    curve: Curve,
     boundary: [(Point<1>, Handle<SurfaceVertex>); 2],
     global_form: Handle<GlobalEdge>,
 }
@@ -53,7 +53,7 @@ pub struct HalfEdge {
 impl HalfEdge {
     /// Create an instance of `HalfEdge`
     pub fn new(
-        curve: SurfacePath,
+        curve: Curve,
         boundary: [(Point<1>, Handle<SurfaceVertex>); 2],
         global_form: Handle<GlobalEdge>,
     ) -> Self {
@@ -65,7 +65,7 @@ impl HalfEdge {
     }
 
     /// Access the curve that defines the half-edge's geometry
-    pub fn curve(&self) -> SurfacePath {
+    pub fn curve(&self) -> Curve {
         self.curve
     }
 
diff --git a/crates/fj-kernel/src/objects/stores.rs b/crates/fj-kernel/src/objects/stores.rs
index 6471fd8e8..7e2272843 100644
--- a/crates/fj-kernel/src/objects/stores.rs
+++ b/crates/fj-kernel/src/objects/stores.rs
@@ -1,7 +1,7 @@
 use fj_math::Vector;
 
 use crate::{
-    geometry::{path::GlobalPath, surface::SurfaceGeometry},
+    geometry::{curve::GlobalPath, surface::SurfaceGeometry},
     storage::{Handle, Store},
 };
 
diff --git a/crates/fj-kernel/src/partial/mod.rs b/crates/fj-kernel/src/partial/mod.rs
index 96b2d0a55..693cfd3e1 100644
--- a/crates/fj-kernel/src/partial/mod.rs
+++ b/crates/fj-kernel/src/partial/mod.rs
@@ -16,7 +16,7 @@ mod wrapper;
 
 pub use self::{
     objects::{
-        curve::MaybeSurfacePath,
+        curve::MaybeCurve,
         cycle::PartialCycle,
         edge::{PartialGlobalEdge, PartialHalfEdge},
         face::PartialFace,
diff --git a/crates/fj-kernel/src/partial/objects/curve.rs b/crates/fj-kernel/src/partial/objects/curve.rs
index f0df764e0..74a26b14a 100644
--- a/crates/fj-kernel/src/partial/objects/curve.rs
+++ b/crates/fj-kernel/src/partial/objects/curve.rs
@@ -1,28 +1,25 @@
 use fj_math::Scalar;
 
-use crate::geometry::path::SurfacePath;
+use crate::geometry::curve::Curve;
 
-/// The definition of a surface path within a partial object
-///
-/// Can be a fully defined [`SurfacePath`], or just the type of path might be
-/// known.
+/// A possibly undefined curve
 #[derive(Clone, Copy, Debug)]
-pub enum MaybeSurfacePath {
-    /// The surface path is fully defined
-    Defined(SurfacePath),
+pub enum MaybeCurve {
+    /// The curve is fully defined
+    Defined(Curve),
 
-    /// The surface path is undefined, but we know it is a circle
+    /// The curve is undefined, but we know it is a circle
     UndefinedCircle {
         /// The radius of the undefined circle
         radius: Scalar,
     },
 
-    /// The surface path is undefined, but we know it is a line
+    /// The curve is undefined, but we know it is a line
     UndefinedLine,
 }
 
-impl From<SurfacePath> for MaybeSurfacePath {
-    fn from(path: SurfacePath) -> Self {
+impl From<Curve> for MaybeCurve {
+    fn from(path: Curve) -> Self {
         Self::Defined(path)
     }
 }
diff --git a/crates/fj-kernel/src/partial/objects/edge.rs b/crates/fj-kernel/src/partial/objects/edge.rs
index 0940a2356..84e201221 100644
--- a/crates/fj-kernel/src/partial/objects/edge.rs
+++ b/crates/fj-kernel/src/partial/objects/edge.rs
@@ -5,7 +5,7 @@ use fj_math::Point;
 
 use crate::{
     objects::{GlobalEdge, GlobalVertex, HalfEdge, Objects, SurfaceVertex},
-    partial::{FullToPartialCache, MaybeSurfacePath, Partial, PartialObject},
+    partial::{FullToPartialCache, MaybeCurve, Partial, PartialObject},
     services::Service,
 };
 
@@ -13,7 +13,7 @@ use crate::{
 #[derive(Clone, Debug)]
 pub struct PartialHalfEdge {
     /// The curve that the half-edge is defined in
-    pub curve: Option<MaybeSurfacePath>,
+    pub curve: Option<MaybeCurve>,
 
     /// The vertices that bound the half-edge on the curve
     pub vertices: [(Option<Point<1>>, Partial<SurfaceVertex>); 2],
@@ -49,7 +49,7 @@ impl PartialObject for PartialHalfEdge {
 
     fn build(self, objects: &mut Service<Objects>) -> Self::Full {
         let curve = match self.curve.expect("Need path to build curve") {
-            MaybeSurfacePath::Defined(path) => path,
+            MaybeCurve::Defined(path) => path,
             undefined => {
                 panic!(
                     "Trying to build curve with undefined path: {undefined:?}"